Theorem finlocfin 23405

Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
finlocfin.1	⊢ 𝑋 = ∪ 𝐽
finlocfin.2	⊢ 𝑌 = ∪ 𝐴

Assertion

Ref	Expression
finlocfin	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Proof of Theorem finlocfin

Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐽 ∈ Top)
2		simp3 1138	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3		simpl1 1192	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Top)
4		finlocfin.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 22791	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6	3, 5	syl 17	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ 𝐽)
7		simpr 484	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8		simpl2 1193	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ Fin)
9		ssrab2 4031	. . . . 5 ⊢ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴
10		ssfi 9087	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
11	8, 9, 10	sylancl 586	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
12		eleq2 2817	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑥 ∈ 𝑛 ↔ 𝑥 ∈ 𝑋))
13		ineq2 4165	. . . . . . . . 9 ⊢ (𝑛 = 𝑋 → (𝑠 ∩ 𝑛) = (𝑠 ∩ 𝑋))
14	13	neeq1d 2984	. . . . . . . 8 ⊢ (𝑛 = 𝑋 → ((𝑠 ∩ 𝑛) ≠ ∅ ↔ (𝑠 ∩ 𝑋) ≠ ∅))
15	14	rabbidv 3402	. . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅})
16	15	eleq1d 2813	. . . . . 6 ⊢ (𝑛 = 𝑋 → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin))
17	12, 16	anbi12d 632	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)))
18	17	rspcev 3577	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
19	6, 7, 11, 18	syl12anc 836	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
20	19	ralrimiva 3121	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
21		finlocfin.2	. . 3 ⊢ 𝑌 = ∪ 𝐴
22	4, 21	islocfin 23402	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
23	1, 2, 20, 22	syl3anbrc 1344	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  ∪ cuni 4858  ‘cfv 6482  Fincfn 8872  Topctop 22778  LocFinclocfin 23389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-om 7800  df-1o 8388  df-en 8873  df-fin 8876  df-top 22779  df-locfin 23392

This theorem is referenced by:  locfincmp  23411  cmppcmp  33841